Given a symmetric, real, Toeplitz matrix:
$$ T=\left( \begin{array}{ccccc} t_1 & t_2 & \ldots & \ldots & t_n \\ t_2 & t_1 & \ddots & \ddots & \vdots \\ \vdots & \ddots & t_1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & t_1 & t_2 \\ t_n & \cdots & \cdots & t_2 & t_1 \\ \end{array} \right) $$
Assuming $n$ is divisible by $2$, I can block-diagonalize this into two blocks via a unitary similarity transform off of:
$$ R=\left( \begin{array}{cccccc} \frac{1}{\sqrt{2}} & & & & & \frac{1}{\sqrt{2}} \\ & \ddots & & & .\cdot{}^{\cdot} & \\ & & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & & \\ & & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & & \\ & .\cdot{}^{\cdot} & & & \ddots & \\ \frac{1}{\sqrt{2}} & & & & & -\frac{1}{\sqrt{2}} \\ \end{array} \right) $$
here's an example of the symmetrization for the $10x10$ case:
$$ {RTR}^\intercal= \\ \left( \begin{array}{cccccccccc} t_1+t_{10} & t_2+t_9 & t_3+t_8 & t_4+t_7 & t_5+t_6 & & & & & \\ t_2+t_9 & t_1+t_8 & t_2+t_7 & t_3+t_6 & t_4+t_5 & & & & & \\ t_3+t_8 & t_2+t_7 & t_1+t_6 & t_2+t_5 & t_3+t_4 & & & & & \\ t_4+t_7 & t_3+t_6 & t_2+t_5 & t_1+t_4 & t_2+t_3 & & & & & \\ t_5+t_6 & t_4+t_5 & t_3+t_4 & t_2+t_3 & t_1+t_2 & & & & & \\ & & & & & t_1-t_2 & t_2-t_3 & t_3-t_4 & t_4-t_5 & t_5-t_6 \\ & & & & & t_2-t_3 & t_1-t_4 & t_2-t_5 & t_3-t_6 & t_4-t_7 \\ & & & & & t_3-t_4 & t_2-t_5 & t_1-t_6 & t_2-t_7 & t_3-t_8 \\ & & & & & t_4-t_5 & t_3-t_6 & t_2-t_7 & t_1-t_8 & t_2-t_9 \\ & & & & & t_5-t_6 & t_4-t_7 & t_3-t_8 & t_2-t_9 & t_1-t_{10} \\ \end{array} \right) $$
We can see we get an additive block and a subtractive block, which in the context this comes from represent a symmetric and anti-symmetric combination of a set of basis functions.
Is there a way to generalize this to $m$ blocks for divisible $n$ and $m$? My intuition suggests it ought to be as the original case derives from effectively a symmetry of order $2$ and there should are symmetries of order $m$ for every $m$, but I don't know where to start in constructing an $R$ for that case.